Introduction to Stan - Moodle@Units · With Stan, we aim to provide an MCMC implementation that...

• What is Stan? • Why Stan? • Writing a Stan program • Linked package: bayesplot •

Bayesian Statistics

Introduction to Stan

Leonardo Egidi

A.A. 2019/20

Leonardo Egidi Introduction 1 / 52


Origins

Stanislaw Ulam (1909-1984): Manhattan project, H-Bomb experiments inLos Alamos, MCMC father jointly with John von Neumann.



Indice

1 What is Stan?

2 Why Stan?

3 Writing a Stan program

4 Linked package: bayesplot



What is Stan?

Probabilistic programming language and inference algorithms.

Stan program

declares data and (constrained) parameter variablesde�nes log posterior (or penalized likelihood)

Stan inference

MCMC for full BayesVariational Bayes for approximate BayesOptimization for (penalized) MLE

Stan ecosystem

lang, math library (C++)interfaces and tools (R, Python, Julia, many more)documentation (example model repo, user guide & reference manual ,

case studies , R package vignettes)online community ( Stan Forums on Discourse)


https://mc-stan.org/users/documentation/

https://mc-stan.org/users/documentation/case-studies.html

https://discourse.mc-stan.org/


Why Stan?

Fit rich Bayesian statistical models. Close to the big data philosophy.

E�ciency

Hamiltonian Monte Carlo + NUTSCompiled to C++

Flexible domain speci�c language

�Freedom-respecting, open-source�

doc & written materialsinteracting communitycontinuous development

Interaction with some other R packages designed to explore the Stanoutput.



Who is using Stan?

Biological & physical sciences: clinical trials, epidemiology, genomics,population ecology, entomology, ophthalmology, neurology, agriculture,�sheries, cancer biology, astrophysics & cosmology, molecular biology,oceanography, climatology.

Social sciences: population dynamics. psycholinguistics, socialnetworks, political science, human development, economics.

Many more: sports analytics, public health, publishing, �nance,pharma, actuarial, recommender systems, educational testing,materials engineering.



Interfaces



Indice

1 What is Stan?

2 Why Stan?





Improving MCMC performance

With Stan, we aim to provide an MCMC implementation that worksrobustly for as many target distributions as possible

Gibbs, RW Metropoilis can be very ine�cient, hard to diagnose.

To explore complicated high-dimensional spaces we need to leveragewhat we know about the geometry of the typical set.

For such a reason, Stan enjoys Hamiltonian Monte Carlo.

We will have now only a brief sketch about how HMC works (furtherreadings are mentioned later on). The Stan users, however, may use,analyze and interpret HMC outputs as they were standard MCMC outputs!



Moving to Hamiltonian Monte Carlo

Once we have built a model, Bayesian computation reduces to evaluatingexpectations, or integrals.

Eπ(θ|y) =∫θπ(θ|y)dθ (1)

How do we compute posterior expectations in practice?

Construct a Markov chain that explores the parameter space.Anything you would want to do if you could write it analytically, youcan do to any accuracy with the draws (history) of the chain

limS→∞

1

S

S∑s=1

θ(s) → Eπ(θ|y)




To be e�cient we need to focus computation on the relevantneighborhoods of parameter space. Relevant neighborhoods, however, arede�ned not by probability density but rather by probability mass.

But exactly which neighborhoods end up contributing most to arbitraryexpectations?

The neighborhoods around the maxima of probability distributions featurea lot of probability density, but, especially in a large number of dimensions,or in long tailed distributions, they do not feature much volume. In otherwords, the sliver size dθ tends to be small there.



The Geometry of High-Dimensional Spaces

Consider a rectangular partitioning centered around a distinguished point,such as the mode (example from Betancourt, 2017):

One of the characteristic properties of high-dimensional spaces is that thereis much more volume outside any given neighborhood than inside of it!



Typical set

Thus, relevant neighborhoods are de�ned not by probability density butrather by probability mass.



Typical set

Probability mass concentrates on a hypersurface called the typical set thatsurrounds the mode.




To accurately estimate expectations we need a method for numerically�nding and then exploring the typical set.

A Markov transition that targets our desired distribution naturallyconcentrates towards probability mass. An inherent ine�ciency in theGibbs sampler and in the random walk Metropolis Hastings is their randomwalk behaviour: the simulations can take a long time zigging and zaggingwhile moving through the target distribution.

HMC borrows strengths from physics to suppress the random walkbehaviour in the Metropolis algorithm, thus allowing it to move much morerapidly through the target distribution. The method enjoys the gradient ofthe log-posterior distribution, d log(π(θ|y))

dθ for a sort of adjustment of thealgorithm towards the typical set area.



When something goes wrong

Under ideal conditions, MCMC estimators converge to the trueexpectations in a very practical progression.












There are many pathological posterior geometries, however, that spoil theseideal conditions.











Hamiltonian Monte Carlo

Hamiltonian Monte Carlo yields fast, and robust, exploration of thedistributions common in practice.



Hamiltonian Monte Carlo: bivariate Gaussian



Indice

1 What is Stan?

2 Why Stan?





Before starting

What is a Bayesian model?

Building a Bayesian model forces us to build a model for how the datais generated

We often think of this as specifying a prior and a likelihood, as if theseare two separate things

They are not!



Generative models

The philosophy behind Stan is to think generatively.

The model is expressed as a joint probability distribution of observed andunobserved variables, which may be decomposed as follows:

p(y , θ) = p(y |θ)π(θ) (2)

The posterior of interest is then proportional to the joint distribution (2):

p(θ|y) ∝ p(y |θ)π(θ) (3)



Generative models

A Bayesian modeller commits to to an a priori joint distribution:

p(y , θ) = p(y |θ)π(θ)︸︷︷︸Likelihood×Prior

= π(θ|y)p(y)︸︷︷︸Posterior×Marginal Likelihood

(4)



Generative models and vague priors

What is the problem with vague/di�use priors?

If we use an improper prior, then we do not specify a joint model forour data and parameters.

More importantly, we do not specify a data generating mechanismp(y).

By construction, these priors do not regularize inferences, which isquite often a bad idea

Proper but di�use is better than .improper but is still oftenproblematic.



Generative models

If we disallow improper priors, then Bayesian modeling is generative.

In particular, we have a simple way to simulate from p(y):



Stan computations

Stan works in logarithmic terms: all the computations are actually done onlog-scale. So, for the posterior we have.

log(π(θ|y)) = log(π(θ)) + log(p(y |θ)) + constant (5)

Products become sums of logs:

p(y |θ) =n∏

i=1

p(yi |θ)→ log(p(y |θ)) =n∑

i=1

log(p(yi |θ)).



Starting point

We are now going to write a Stan program together:

Open a new empty �le in RStudio

Save it as linear_regression.stan



Blocks strategy

Stan programs are organized into blocks:

data block: declare data types, sizes, and constraints. Read from datasource and constraints validated. Evaluated: once.

parameters block: declare parameter types, sizes, and constraints.Evaluated: every log prob evaluation.

transformed parameters block: declare those parameters transformedfrom the original ones declared in the parameters block. Evaluated:every log prob evaluation.

model block: statements de�ning the posterior density in log scale.Evaluated: every log prob evaluation.

generated quantities: declare and de�ne derived variables. (P)RNGs,predictions, event probabilities, decision making. Constraintsvalidated. Evaluated: once per draw.



Data block



Parameters' block



Model block



Generated quantities block



Complete Stan model



Launching the Stan model from R

Now we may launch the Stan program directly in R:

library(rstan)

# passing the data (already stored)

data <- list(N=N, K=K, X=X, y=y)

# fitting the model

fit1 <- stan(

file = 'linear_regression.stan',

data = data,

iter = 2000,

chains = 4)

# extracting the estimates

sims <- extract(fit1)



First example: 8 schools

This example studied coaching e�ects from eight schools.We denote with yij the result of the i-th test in the j-th school. We assumethe following model:

yij ∼ N (θj , σ2

y )

θj ∼ N (µ, τ2)

Do some schools perform better/worse according to these coaching e�ects?Here is the data, already aggregated by schools:

schools_dat <- list(J = 8,

y = c(28, 8, -3, 7, -1, 1, 18, 12),

sigma = c(15, 10, 16, 11, 9, 11, 10, 18))



First Stan model: 8 schools

// saved as 8schools.stan

data {

int<lower=0> J; // number of schools

real y[J]; // estimated treatment effects

real<lower=0> sigma[J]; // standard error of effect estimates

}

parameters {

real mu; // population treatment effect

real<lower=0> tau; // standard deviation in treatment effects

vector[J] eta; // unscaled deviation from mu by school

}

transformed parameters {

vector[J] theta = mu + tau * eta; // school treatment effects

}

model {

eta ~ normal(0,1); // prior

y ~ normal(theta, sigma); //likelihood

}Leonardo Egidi Introduction 40 / 52


First example: 8 schools

To �t the model and visualize the estimates, it is su�cient to type in R thefollowing commands (with 2000 iterations and 4 chains as a default):

fit_8schools <- stan(file = '8schools.stan', data = schools_dat)

print(fit_8schools, pars=c("mu", "tau", "theta"))

mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat

mu 7.89 5.04 -2.31 4.74 7.92 11.05 18.05 2352 1

tau 6.70 5.71 0.24 2.61 5.43 9.19 21.16 1480 1

theta[1] 11.36 8.23 -2.25 6.18 10.29 15.46 31.15 3161 1

theta[2] 7.89 6.21 -4.43 3.96 7.83 11.78 20.47 4923 1

theta[3] 6.05 7.59 -10.81 1.92 6.56 10.81 20.25 4057 1

theta[4] 7.60 6.44 -5.36 3.74 7.63 11.73 20.57 5055 1

theta[5] 5.13 6.23 -8.45 1.35 5.60 9.26 16.37 4346 1

theta[6] 5.95 6.68 -8.21 1.99 6.30 10.21 18.21 4313 1

theta[7] 10.62 6.93 -1.58 6.12 10.14 14.53 25.63 3381 1

theta[8] 8.40 7.77 -7.18 3.84 8.26 12.63 25.78 3854 1



Indice

1 What is Stan?

2 Why Stan?





Posterior graphical analysis with bayesplot

Once we �t a model, it is to vital check it via graphical inspection. Thebayesplot package (for any help, see the vignette ) is designed to this task.

The package allows to display:

Posterior uncertainty intervals

Univariate marginal posterior distributions

Bivariate plots

Trace plots

Posterior predictive plots


https://cran.r-project.org/web/packages/bayesplot/vignettes/plotting-mcmc-draws.html


Posterior graphical analysis with bayesplot

The �rst step is to save the posterior. Then you have many choices:

library(bayesplot)

posterior <- as.array(fit_8schools)

mcmc_intervals(posterior) # posterior intervals

mcmc_areas(posterior) # posterior areas

mcmc_dens(posterior) # marginal posteriors

mcmc_pairs(posterior) # bivariate plots

mcmc_trace(posterior) # trace plots

With the arguments pars or regex_pars you may select the desiredparameters.



Posterior uncertainty intervals

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θ1

θ2

θ3

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τ

µ

−10 0 10 20 30

Posterior intervals



Posterior uncertainty areas

θ1

θ2

θ3

θ4

θ5

θ6

θ7

θ8

τ

µ

−25 0 25 50 75

Posterior areas



Marginal posteriors

theta[6] theta[7] theta[8] lp__

theta[1] theta[2] theta[3] theta[4] theta[5]

eta[4] eta[5] eta[6] eta[7] eta[8]

mu tau eta[1] eta[2] eta[3]

−20 0 20 0 20 40 −20 0 20 40 60 −15 −10 −5 0

0 20 40 60 −10 0 10 20 30 −20 0 20 40 0 20 40 −20 −10 0 10 20

−2 0 2 −4 −2 0 2 −3 −2 −1 0 1 2 3 −2 0 2 −4 −2 0 2

−10 0 10 20 30 20 40 60 −2 0 2 −2 0 2 −2 0 2



Marginal posteriors separated for each chain





−20 0 20 0 20 40 −20 0 20 40 60 −15 −10 −5 0

0 20 40 60 −10 0 10 20 30 −20 0 20 40 0 20 40 −20 −10 0 10 20

−2 0 2 −4 −2 0 2 −3 −2 −1 0 1 2 3 −2 0 2 −4 −2 0 2

−10 0 10 20 30 20 40 60 −2 0 2 −2 0 2 −2 0 2

Chain

1234



Bivariate posterior plots



Trace plots for the Markov chains





0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000

0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000

0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000

0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000−4

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Chain

1234



Our challenge with Stan

The Stan shuttle is ready to start! We will learn to:

write simple and more complex model in Stan: lm, glm, hierarchicalmodels.

analyze the posterior summaries.

criticize the model and, eventually, change/reparametrize it.



Further reading

Further reading:

Carpenter, B, and Gelman, A, Ho�man, M.D., Lee, D., Goodrich, B.,Betancourt, M., Brubaker, M., Guo, J., Li, P., Riddell, A. (2017).Stan: A Probabilistic Programming Language, Journal of statisticalsoftware 76(1). Here the pdf

Further optional reading about Hamiltonian Monte Carlo:

Betancourt, M. (2017) A conceptual introduction to HamiltonianMonte Carlo. Here the pdf


https://www.osti.gov/servlets/purl/1430202

https://arxiv.org/pdf/1701.02434.pdf

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